In mathematics, the Pettis integral or Gelfand–Pettis integral, named after Israel M. Gelfand and Billy James Pettis, extends the definition of the Lebesgue integral to vector-valued functions on a measure space, by exploiting duality. The integral was introduced by Gelfand for the case when the measure space is an interval with Lebesgue measure. The integral is also called the weak integral in contrast to the Bochner integral, which is the strong integral.
Definition Let f : X → V {\displaystyle f:X\to V} where ( X , Σ , μ ) {\displaystyle (X,\Sigma ,\mu )} is a measure space and V {\displaystyle V} is a topological vector space (TVS) with a continuous dual space V ′ {\displaystyle V'} that separates points (that is, if x ∈ V {\displaystyle x\in V} is nonzero then there is some l ∈ V ′ {\displaystyle l\in V'} such that l ( x ) ≠ 0 {\displaystyle l(x)\neq 0} ), for example, V {\displaystyle V} is a normed space or (more generally) is a Hausdorff locally convex TVS. Evaluation of a functional may be written as a duality pairing:
⟨ φ , x ⟩ = φ [ x ] . {\displaystyle \langle \varphi ,x\rangle =\varphi [x].}
The map f : X → V {\displaystyle f:X\to V} is called weakly measurable if for all φ ∈ V ′ , {\displaystyle \varphi \in V',} the scalar-valued map φ ∘ f {\displaystyle \varphi \circ f} is a measurable map. A weakly measurable map f : X → V {\displaystyle f:X\to V} is said to be weakly integrable on X {\displaystyle X} if there exists some e ∈ V {\displaystyle e\in V} such that for all φ ∈ V ′ , {\displaystyle \varphi \in V',} the scalar-valued map φ ∘ f {\displaystyle \varphi \circ f} is Lebesgue integrable (that is, φ ∘ f ∈ L 1 ( X , Σ , μ ) {\displaystyle \varphi \circ f\in L^{1}\left(X,\Sigma ,\mu \right)} ) and
φ ( e ) = ∫ X φ ( f ( x ) ) d μ ( x ) . {\displaystyle \varphi (e)=\int _{X}\varphi (f(x))\,\mathrm {d} \mu (x).} The map f : X → V {\displaystyle f:X\to V} is said to be Pettis integrable if φ ∘ f ∈ L 1 ( X , Σ , μ ) {\displaystyle \varphi \circ f\in L^{1}\left(X,\Sigma ,\mu \right)} for all φ ∈ V ′ {\displaystyle \varphi \in V^{\prime }} and also for every A ∈ Σ {\displaystyle A\in \Sigma } there exists a vector e A ∈ V {\displaystyle e_{A}\in V} such that
⟨ φ , e A ⟩ = ∫ A ⟨ φ , f ( x ) ⟩ d μ ( x ) for all φ ∈ V ′ . {\displaystyle \langle \varphi ,e_{A}\rangle =\int _{A}\langle \varphi ,f(x)\rangle \,\mathrm {d} \mu (x)\quad {\text{ for all }}\varphi \in V'.}
In this case, e A {\displaystyle e_{A}} is called the Pettis integral of f {\displaystyle f} on A . {\displaystyle A.} Common notations for the Pettis integral e A {\displaystyle e_{A}} include
∫ A f d μ , ∫ A f ( x ) d μ ( x ) , and, in case that A = X is understood, μ [ f ] . {\displaystyle \int _{A}f\,\mathrm {d} \mu ,\qquad \int _{A}f(x)\,\mathrm {d} \mu (x),\quad {\text{and, in case that}}~A=X~{\text{is understood,}}\quad \mu [f].}
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